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%I #13 Dec 07 2020 08:16:39
%S 0,0,1,1,3,4,8,11,19,26,41,56,83,112,159,211,291,381,512,663,873,1117,
%T 1448,1833,2342,2938,3708,4611,5760,7105,8792,10769,13215,16077,19585,
%U 23679,28651,34447,41424,49541,59248,70509,83892,99390,117695,138846,163708
%N Number of partitions of n containing at least one part m-10 if m is the largest part.
%H Alois P. Heinz, <a href="/A212550/b212550.txt">Table of n, a(n) for n = 10..1000</a>
%F G.f.: Sum_{i>0} x^(2*i+10) / Product_{j=1..10+i} (1-x^j).
%e a(12) = 1: [11,1].
%e a(13) = 1: [11,1,1].
%e a(14) = 3: [11,1,1,1], [11,2,1], [12,2].
%e a(15) = 4: [11,1,1,1,1], [11,2,1,1], [11,3,1], [12,2,1].
%e a(16) = 8: [11,1,1,1,1,1], [11,2,1,1,1], [11,2,2,1], [11,3,1,1], [11,4,1], [12,2,1,1], [12,2,2], [13,3].
%p b:= proc(n, i) option remember;
%p `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p end:
%p a:= n-> add(b(n-2*m-10, min(n-2*m-10, m+10)), m=1..(n-10)/2):
%p seq(a(n), n=10..60);
%t Table[Count[IntegerPartitions[n],_?(MemberQ[#,#[[1]]-10]&)],{n,10,60}] (* _Harvey P. Dale_, Feb 10 2015 *)
%t b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];
%t a[n_] := Sum[b[n - 2m - 10, Min[n - 2m - 10, m + 10]], {m, 1, (n - 10)/2}];
%t a /@ Range[10, 60] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y Column k=10 of A212551.
%K nonn
%O 10,5
%A _Alois P. Heinz_, May 20 2012
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